Algebra Matrix Inverse: Expressing x variables in terms of z variables"

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In summary, the question asks to express the variables x1, x2, and x3 in terms of the variables z1, z2, and z3 given two matrices. The equations for x1, x2, and x3 are found by multiplying the first matrix by the variables y1, y2, and y3. The equations for z1, z2, and z3 are found by multiplying the second matrix by the same variables. To solve the problem, the inverse of the product of the two matrices must be found or the inverses of each matrix must be multiplied.
  • #1
vg19
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Hey again!
Im having trouble with this problem given in the matrix inverse section of the textbook. It gives these two matricies in the form AX=B
[x1]=[3 -1 2][y1]
[x2]=[1 0 4][y2]
[x3]=[2 1 0][y3]
and
[z1]=[1 -1 1][y1]
[z2]=[2 -3 0][y2]
[z3]=[-1 1 -2][y3]
The question says, given the first matrix and the second matrix, express the variables, x1, x2, x3 in terms of z1, z2, z3. I am not too sure on where to start here. So far, I just multiplied through to find the equations for the x variables and z variables.
x1 = 3y1 - y2 + 2y3
x2 = y1 + 4y3
x2 = 2y1 + y2
z1 = y1 - y2 + y3
z2 = 2y1 - 3y2
z3 = -y1 + y2 + 2y3
Im not sure on where to go from here.
Thanks in advance
 
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  • #2
If Y= AX and Z= BY then Z=B(AX)= (BA)X and so X= (BA)-1Z= (A-1B-1)Z. Can you find the inverses of those matrices?
(Or multiply them and then find the inverse of the product.)
 

FAQ: Algebra Matrix Inverse: Expressing x variables in terms of z variables"

1. What is an algebra matrix inverse?

An algebra matrix inverse is the matrix that, when multiplied by the original matrix, results in the identity matrix. This means that the inverse matrix "undoes" the original matrix, similar to how division "undoes" multiplication.

2. Why is it important to express x variables in terms of z variables?

Expressing x variables in terms of z variables allows us to solve for specific variables in a system of equations. It also helps us understand the relationships between variables and how changes in one variable affect the others.

3. How do you find the inverse of a matrix?

To find the inverse of a matrix, you can use the Gauss-Jordan elimination method or use the formula (1/det(A)) * adj(A), where det(A) is the determinant of the matrix and adj(A) is the adjugate matrix (transpose of the cofactor matrix).

4. Can any matrix have an inverse?

No, not all matrices have an inverse. A matrix must be square and have a nonzero determinant in order to have an inverse.

5. Why is it important to check if a matrix has an inverse before finding it?

It is important to check if a matrix has an inverse before finding it because if a matrix does not have an inverse, attempting to find it will result in an error. Additionally, knowing if a matrix has an inverse can also help us determine if a system of equations has a unique solution or not.

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